(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 12.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 63756, 1251] NotebookOptionsPosition[ 61774, 1217] NotebookOutlinePosition[ 62253, 1235] CellTagsIndexPosition[ 62210, 1232] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[TextData[{ "F. Fass\[OGrave]. University of Padova. \n", StyleBox["Introduction to the use of Mathematica in Mathematics and Science", FontSlant->"Italic"], ". SoftSkills course for the PhD program in Mathematics\nExercise for the \ first three lectures (October 28, 2023)" }], "Text", CellFrame->{{0, 0}, {0.5, 0}}, CellChangeTimes->{{3.6030403380477047`*^9, 3.6030403496377206`*^9}, { 3.6042342921245947`*^9, 3.604234293844597*^9}, {3.6659842250605717`*^9, 3.665984234810585*^9}, {3.665988776604652*^9, 3.6659887774046535`*^9}, { 3.666106805730517*^9, 3.666106807983646*^9}, {3.686715090311494*^9, 3.68671511399888*^9}, {3.6973875672817383`*^9, 3.6973875784126663`*^9}, { 3.697434883527854*^9, 3.697434886700633*^9}, {3.7288472476388865`*^9, 3.728847250442567*^9}, 3.7290563713306975`*^9, {3.7603265886670856`*^9, 3.760326590879668*^9}, {3.792953448635523*^9, 3.7929534512106314`*^9}, { 3.793001443067047*^9, 3.793001457902358*^9}, {3.793125487673247*^9, 3.7931254933125515`*^9}, {3.823794609368146*^9, 3.8237946203338003`*^9}, { 3.905514199116931*^9, 3.90551422872608*^9}, {3.9055144580037365`*^9, 3.9055145289462028`*^9}, 3.905524053605975*^9, {3.9056541789884577`*^9, 3.90565419524717*^9}, {3.906167256174015*^9, 3.9061672588858814`*^9}, { 3.9067247773353424`*^9, 3.906724780763774*^9}, {3.9074107712454653`*^9, 3.9074107934173474`*^9}, {3.907507135155755*^9, 3.9075071356480827`*^9}}, FontSize->12, Background->None,ExpressionUUID->"b016031e-6c8e-4a66-ac65-1747af6335ee"], Cell[CellGroupData[{ Cell["Exercises for lectures 1-3", "Title", CellChangeTimes->{{3.6661072320849032`*^9, 3.6661072427585135`*^9}, { 3.6661104489970016`*^9, 3.66611045504701*^9}, {3.66685386493797*^9, 3.6668538678979745`*^9}, {3.729603406077321*^9, 3.729603410616357*^9}, { 3.9074107978591924`*^9, 3.9074108114308844`*^9}, {3.9074113340919065`*^9, 3.9074113346333585`*^9}},ExpressionUUID->"f729008c-babd-4da4-bc56-\ be901629beba"], Cell["\<\ 1. Create a list of the squares of the natural numbers between 1 and 99. \ Extract its 11th element. \ \>", "Text", CellChangeTimes->{{3.666107539131465*^9, 3.666107601891055*^9}, { 3.666108241794655*^9, 3.6661082624828386`*^9}, 3.666108368004874*^9, 3.6661085551218023`*^9, {3.697602269802958*^9, 3.6976022701192026`*^9}, { 3.9074108215034533`*^9, 3.9074108771643515`*^9}}, Background->RGBColor[ 0., 1., 0.],ExpressionUUID->"60ed2545-9357-4dd9-9a2f-7ac787a18d14"], Cell[TextData[{ "2. Define a function f(n) which produces the approximation of ", Cell[BoxData[ FormBox[ SqrtBox["3"], TraditionalForm]],ExpressionUUID-> "4186818a-c307-4525-8861-61431116cbd8"], "-1 with n decimal digits." }], "Text", CellChangeTimes->{{3.6661072470917616`*^9, 3.6661072500919333`*^9}, { 3.6661076370720673`*^9, 3.6661076375670958`*^9}, {3.6661076714910355`*^9, 3.66610772121588*^9}, {3.666107774232912*^9, 3.6661078209845862`*^9}, { 3.69760227195304*^9, 3.6976022881977787`*^9}, {3.9074109120282073`*^9, 3.907410960601859*^9}}, Background->RGBColor[ 0., 1., 0.],ExpressionUUID->"64e41634-1614-4033-8581-bd92fcdf0616"], Cell[TextData[{ "3. Determine the zeroes of the polynomial ", Cell[BoxData[ RowBox[{ SuperscriptBox["x", "3"], "-", RowBox[{"2", "x"}], "+", "1"}]],ExpressionUUID-> "b0b7fa14-c917-4bd4-b3df-f403ecac4d7a"] }], "Text", CellChangeTimes->{{3.666107723280998*^9, 3.666107724177049*^9}, { 3.666107836467472*^9, 3.666107836841493*^9}, {3.666108570021824*^9, 3.666108585521845*^9}, {3.6661088578932266`*^9, 3.6661088884032693`*^9}, { 3.6661089621033726`*^9, 3.666108986183406*^9}, {3.697602289931883*^9, 3.697602290363816*^9}, {3.907410963520753*^9, 3.9074109789791517`*^9}}, Background->RGBColor[ 0., 1., 0.],ExpressionUUID->"f994dc9a-238f-4cb5-b156-e4f1eaa9a8d4"], Cell[TextData[{ "4. Compute the length of the ellipse of semiaxes a=2 and b=1? \n\ [Suggestions: a parametrization of the ellipse is\n t \[Rule] (a \ cos(t) , b sin(t) ) , 0\[LessEqual] t\[LessEqual]2\[Pi] .\nThe length of a \ curve t\[Rule] \[Gamma](t) = (x(t),y(t)) is the integral of ||\[Gamma]\ \[CloseCurlyQuote](t)|| = ", Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{ RowBox[{ RowBox[{"x", "'"}], SuperscriptBox[ RowBox[{"(", "t", ")"}], "2"]}], "+", RowBox[{ RowBox[{"y", "'"}], SuperscriptBox[ RowBox[{"(", "t", ")"}], "2"]}]}]], TraditionalForm]],ExpressionUUID-> "faf74e18-996a-403b-b30f-cec26334b15b"], " . See the help for how to compute a definte integral. You can compute the \ integral symbolically of numerically. " }], "Text", CellChangeTimes->{{3.666107723280998*^9, 3.666107724177049*^9}, { 3.666107836467472*^9, 3.666107836841493*^9}, {3.666108570021824*^9, 3.666108585521845*^9}, {3.6661088578932266`*^9, 3.6661088884032693`*^9}, { 3.6661089621033726`*^9, 3.666108986183406*^9}, {3.6668569063552284`*^9, 3.6668570030553637`*^9}, {3.666857041125417*^9, 3.666857069965458*^9}, { 3.6668571864956207`*^9, 3.66685725711572*^9}, {3.66685744302598*^9, 3.6668575718261604`*^9}, {3.6976018754098597`*^9, 3.6976020783986807`*^9}, { 3.697602301894182*^9, 3.69760230231605*^9}, {3.729603290724444*^9, 3.729603297536945*^9}, {3.729605261108821*^9, 3.729605325038097*^9}, { 3.7937161154234533`*^9, 3.793716115845219*^9}, {3.794981974750619*^9, 3.794981998522052*^9}, {3.907410982371433*^9, 3.9074112500408936`*^9}}, Background->RGBColor[ 0., 1., 0.],ExpressionUUID->"6827507b-dd8b-4b01-b59f-dc593c3b7887"], Cell[TextData[{ "5. Compute the area of an ellipse of semiaxes a and b. \ \[LineSeparator][Suggestions: if the Cartesian equation of the ellipse is ", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ FractionBox[ SuperscriptBox["x", "2"], SuperscriptBox["a", "2"]], "+", " ", FractionBox[ SuperscriptBox["y", "2"], SuperscriptBox["b", "2"]]}]}], TraditionalForm]],ExpressionUUID-> "1a9f47b8-981e-407f-b12c-3c56cbd03a14"], "=1, the area is twice the integral of ", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{"b", SqrtBox[ RowBox[{"(", RowBox[{"1", "-", " ", FractionBox[ SuperscriptBox["x", "2"], SuperscriptBox["a", "2"]]}], ")"}]]}]}], TraditionalForm]], ExpressionUUID->"fd59568a-e678-407e-9d70-edbd6c2ff6cd"], " for - ", Cell[BoxData[ FormBox["a", TraditionalForm]], FormatType->TraditionalForm,ExpressionUUID-> "cc9bcf2d-241d-4cb5-9415-87f1496150f4"], " \[LessEqual] x \[LessEqual] ", Cell[BoxData[ FormBox["a", TraditionalForm]], FormatType->TraditionalForm,ExpressionUUID-> "d3bd584e-f065-4b58-a5a6-b40f6a79ed2a"], " ].\nWrite now the equation of the ellipse as ", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{"A", " ", SuperscriptBox["x", "2"]}], "+", " ", RowBox[{"B", " ", SuperscriptBox["y", "2"]}]}]}], TraditionalForm]],ExpressionUUID-> "48494513-983f-4b9d-b2f4-f7ae3949dace"], "=1 and the area as twice the integral of ", Cell[BoxData[ FormBox[ RowBox[{" ", SqrtBox[ RowBox[{ FractionBox[ RowBox[{"1", "-", " ", SuperscriptBox["Ax", "2"]}], "B"], ")"}]]}], TraditionalForm]], ExpressionUUID->"c52c27ae-aa8f-4d3f-aa83-d6fe2964751b"], " for - ", Cell[BoxData[ FormBox[ FractionBox["1", SqrtBox["A"]], TraditionalForm]], FormatType->TraditionalForm,ExpressionUUID-> "5f610746-1697-4196-994b-739e84b553f5"], " \[LessEqual] x \[LessEqual] ", Cell[BoxData[ FormBox[ FractionBox["1", SqrtBox["A"]], TraditionalForm]],ExpressionUUID-> "c1e0fea4-6bba-468a-a8cf-05c8b71b79d9"], " ].\nWhy ", StyleBox["Mathematica ", FontSlant->"Italic"], " does not give ", Cell[BoxData[ FormBox[ FractionBox["\[Pi]", SqrtBox["AB"]], TraditionalForm]], FormatType->TraditionalForm,ExpressionUUID-> "4b9441bb-a8fd-425b-b629-de9e96619907"], " as result? Can you \[OpenCurlyDoubleQuote]Refine\[CloseCurlyDoubleQuote] \ it with suitable assumtions? \n[Suggestions: logical AND is written \ \[OpenCurlyDoubleQuote]&&\[CloseCurlyDoubleQuote]]" }], "Text", CellChangeTimes->CompressedData[" 1:eJwdxU0ogwEABuA1KwuXJdJCtjmyNEOyw77247DMTC2pLfMXOYgDOeCwTWtm IUM2qx0wS2QOKDnQakNZ2QFJs/x/W61MGOJ7d3h6OO396i46jUZjU3Dit2d9 UkUSHV5TevO7bRcb4kvpAz5xrpU66avIwyq7qAbbn3LqcFgpkeKhW1k9ZuoU I19NJGHx6024L6PUgpkPh1O45KbRiVM7vBUcDrcd4J/Xj3s8dulOYVvVACNF Xcyn8/Amu5OPdVGhf9lKEr3ZCyd4boB2jrmVsgv8+SaN4I1ZZfpk8EW7FiSJ UUWWDk8bRE6c32Bx4e5al9ZDLS2q1mPmRGQwIYkRXmMivcDhsLXKYwQrZZ3B 86K71dPCOFGuYXqws3l/G4uN40c4cysYxFGWN4QfDcIrrIntRbFc8PeM1S1a Eg+bj5M4QP8uOKMuW7zm4BDDzcX/lXDhKA== "], Background->RGBColor[ 0., 1., 0.],ExpressionUUID->"385a656a-334e-4f7b-bfcb-d56e600d5e61"], Cell[TextData[{ "6. Write a function that creates the list of the first ", StyleBox["n", FontSlant->"Italic"], " odd naturales. Plot the graph of such a list for ", StyleBox["n", FontSlant->"Italic"], "=11, joining the points with segments.\n[Suggestions: ListPlot; Option: \ Joined]" }], "Text", CellChangeTimes->{{3.6661072470917616`*^9, 3.6661072923083477`*^9}, { 3.6661073263872967`*^9, 3.666107393354127*^9}, {3.666107424937934*^9, 3.666107437098629*^9}, {3.666107524195611*^9, 3.666107535230242*^9}, { 3.666854418179745*^9, 3.666854532359905*^9}, {3.7296035450976086`*^9, 3.7296035457327166`*^9}, {3.7296038554112453`*^9, 3.729603858241037*^9}, { 3.729605409910498*^9, 3.7296054352170267`*^9}, {3.7296054691238947`*^9, 3.7296054700700674`*^9}, {3.7937161309666853`*^9, 3.793716131794722*^9}, { 3.9074122557413764`*^9, 3.907412284501383*^9}, {3.9074123300574265`*^9, 3.907412401319829*^9}}, Background->RGBColor[ 0., 1., 0.],ExpressionUUID->"46d707c5-5062-4896-8900-bf26ee63f539"], Cell[TextData[{ "7. Write, in three different ways, a function f(a,b) that plots an ellipse \ of semiaxes a and b: \n - with the ellipse given as a parametrized curve \ in ", Cell[BoxData[ FormBox[ SuperscriptBox[ TemplateBox[{}, "Reals"], "2"], TraditionalForm]], FormatType->TraditionalForm,ExpressionUUID-> "6a7700a1-8927-41b7-aaf2-d0fd51870e30"], "\n - with the ellipse given as a level curve of function ", Cell[BoxData[ FormBox[ SuperscriptBox[ TemplateBox[{}, "Reals"], "2"], TraditionalForm]], FormatType->TraditionalForm,ExpressionUUID-> "3034b184-9e65-475b-a9e0-696ea1043e38"], "\[Rule] ", Cell[BoxData[ TemplateBox[{}, "Reals"]],ExpressionUUID->"10ca22d0-0d69-4a92-a29a-5b0392b9a6ec"], "\n - with the ellipse formed by the union of the graphs of two functions \ ", Cell[BoxData[ FormBox[ RowBox[{"x", "\[Rule]", SubscriptBox["y", "\[PlusMinus]"]}], TraditionalForm]],ExpressionUUID-> "36ed2050-1658-44d4-9263-49fd419ecbb7"], "(x) = \[PlusMinus]", Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"...", "."}]], TraditionalForm]],ExpressionUUID-> "9a934c9f-c241-4344-b22d-a7b4f3b6ff63"], " \n" }], "Text", CellChangeTimes->{{3.666107723280998*^9, 3.666107724177049*^9}, { 3.666107836467472*^9, 3.666107836841493*^9}, {3.666108570021824*^9, 3.666108585521845*^9}, {3.6661088578932266`*^9, 3.6661088884032693`*^9}, { 3.6661089621033726`*^9, 3.666108986183406*^9}, {3.6668569063552284`*^9, 3.6668570030553637`*^9}, {3.6668581976480365`*^9, 3.6668582004480405`*^9}, {3.6668582458381042`*^9, 3.6668582515381117`*^9}, 3.6981571377670307`*^9, 3.698157221197792*^9, {3.7296035770729637`*^9, 3.7296035774470882`*^9}, {3.729603711082547*^9, 3.729603717297896*^9}, { 3.729605904524935*^9, 3.7296059292565937`*^9}, {3.7296059603620806`*^9, 3.729605983597072*^9}, {3.793716137761944*^9, 3.793716138542513*^9}, { 3.7949820408842716`*^9, 3.7949820448051453`*^9}, {3.9074124054406004`*^9, 3.9074126857692013`*^9}}, Background->RGBColor[ 0., 1., 0.],ExpressionUUID->"2e3bcb28-f8a8-49cd-b557-d36ec4bdb18c"], Cell["\<\ 8. Plot a sphere: - as a parametrized surface - as level set of a function\ \>", "Text", CellChangeTimes->{{3.666107723280998*^9, 3.666107724177049*^9}, { 3.666107836467472*^9, 3.666107836841493*^9}, {3.666108570021824*^9, 3.666108585521845*^9}, {3.6661088578932266`*^9, 3.6661088884032693`*^9}, { 3.6661089621033726`*^9, 3.666108986183406*^9}, {3.6668569063552284`*^9, 3.6668570030553637`*^9}, {3.6668581976480365`*^9, 3.6668582004480405`*^9}, {3.6668582458381042`*^9, 3.6668582515381117`*^9}, 3.6981571377670307`*^9, 3.698157221197792*^9, {3.7296035770729637`*^9, 3.7296035774470882`*^9}, {3.729603711082547*^9, 3.729603717297896*^9}, { 3.729605904524935*^9, 3.7296059292565937`*^9}, {3.7296059603620806`*^9, 3.729605983597072*^9}, {3.793716137761944*^9, 3.79371621837422*^9}, { 3.9074126887929325`*^9, 3.9074127266786766`*^9}}, Background->RGBColor[ 0., 1., 0.],ExpressionUUID->"a5f7de18-7674-4454-9e91-07cd96288f80"], Cell[TextData[{ "9. A parametrization of the Moebius band a surface \[Subset] ", Cell[BoxData[ FormBox[ SuperscriptBox[ TemplateBox[{}, "Reals"], "3"], TraditionalForm]],ExpressionUUID-> "b8ed1cd9-83eb-4a7a-a77f-c9e2a4ca01cc"], " is\n\n", Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{"s", ",", "\[CurlyPhi]"}], ")"}], "\[Rule]", " ", RowBox[{ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{"Cos", "[", "\[CurlyPhi]", "]"}], RowBox[{"-", RowBox[{"Sin", "[", "\[CurlyPhi]", "]"}]}], "0"}, { RowBox[{"Sin", "[", "\[CurlyPhi]", "]"}], RowBox[{"Cos", "[", "\[CurlyPhi]", "]"}], "0"}, {"0", "0", "1"} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]], TagBox[ RowBox[{"(", "\[NoBreak]", TagBox[GridBox[{ { RowBox[{"R", "+", RowBox[{"s", " ", RowBox[{"Cos", "[", FractionBox["\[CurlyPhi]", "2"], "]"}]}]}]}, {"0"}, { RowBox[{"s", " ", RowBox[{"Sin", "[", FractionBox["\[CurlyPhi]", "2"], "]"}]}]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.5599999999999999]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], Column], "\[NoBreak]", ")"}], Function[BoxForm`e$, MatrixForm[BoxForm`e$]]]}]}]],ExpressionUUID-> "3a437c92-fcdb-405f-b7e3-c7539f41be42"], " , s\[Element] [-r,r] , \[CurlyPhi] \[Element] [0,2\[Pi]] \n\nwhere R \ > r > 0 are two parameters. 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Find the ", StyleBox["integer", FontSlant->"Italic"], " zeroes of the polynomial ", Cell[BoxData[ RowBox[{"2", " ", SuperscriptBox["x", "2"]}]], "Input",ExpressionUUID-> "a55c7d75-5c5b-47be-b0b3-5c9ce630d885"], " - 5 x - 3\n\nSuggestions:Solve[ equation, variable, domain]" }], "Text", CellChangeTimes->{{3.447484168796875*^9, 3.44748424740625*^9}, { 3.447484302515625*^9, 3.44748434090625*^9}, {3.44748440853125*^9, 3.447484408640625*^9}, {3.4474844545*^9, 3.447484505296875*^9}, { 3.44748467215625*^9, 3.447484709484375*^9}, {3.447485266953125*^9, 3.447485347484375*^9}, {3.447485393234375*^9, 3.44748539540625*^9}, { 3.44748566625*^9, 3.44748566825*^9}, {3.44750038696875*^9, 3.44750044678125*^9}, {3.447502224125*^9, 3.44750223*^9}, { 3.447502313296875*^9, 3.447502380703125*^9}, {3.4475024481875*^9, 3.44750253865625*^9}, {3.447502580875*^9, 3.447502749203125*^9}, { 3.447502805640625*^9, 3.447503103859375*^9}, {3.447511491796875*^9, 3.447511491796875*^9}, {3.447511684796875*^9, 3.44751178165625*^9}, { 3.447585324046875*^9, 3.44758532784375*^9}, {3.45001595128125*^9, 3.4500160753125*^9}, {3.450016107109375*^9, 3.45001626103125*^9}, 3.450016396109375*^9, {3.793716245636414*^9, 3.7937162656415954`*^9}, { 3.7937163062572803`*^9, 3.7937163372728167`*^9}, {3.793717903390607*^9, 3.7937179562114754`*^9}, {3.793718064926051*^9, 3.7937182172990417`*^9}, { 3.79371916283041*^9, 3.7937191953987103`*^9}, {3.793719917093993*^9, 3.7937199747031097`*^9}, {3.7937204000419245`*^9, 3.793720418936866*^9}, { 3.7937204681563735`*^9, 3.793720469833965*^9}, {3.793720787418187*^9, 3.7937207880837507`*^9}, {3.79372088852099*^9, 3.7937208889251003`*^9}, { 3.907412840883844*^9, 3.9074128584218225`*^9}, {3.907412903978726*^9, 3.9074129780819716`*^9}, {3.907413653672807*^9, 3.9074137300451646`*^9}, { 3.9074258998598213`*^9, 3.9074259102075615`*^9}}, Background->RGBColor[ 0., 1., 0.],ExpressionUUID->"edf53c7e-f392-403c-8d41-e2875e7eabf3"], Cell[TextData[{ "12. The \[OpenCurlyDoubleQuote]Number Prime Theorem\[CloseCurlyDoubleQuote] \ states the number \[Pi](n) of primes \[InvisibleComma] \[LessEqual] n is \ asymptotic to ", Cell[BoxData[ FormBox[ FractionBox["n", RowBox[{"log", "(", "n", ")"}]], TraditionalForm]],ExpressionUUID-> "7cdfa006-ff7f-4743-8e3d-3c8e88a892a1"], " for n\[Rule]\[Infinity].\n\n- In ", StyleBox["Mathematica", FontSlant->"Italic"], ", \[Pi] is the function PrimePi, and is defined for real arguments.\nPlot \ together the two functions ", Cell[BoxData[ FormBox[ RowBox[{"\[Pi]", "(", "x", ")"}], TraditionalForm]],ExpressionUUID-> "8d7d8497-9a12-41a1-a22a-0ede8521cec3"], " and ", Cell[BoxData[ FormBox[ FractionBox["x", RowBox[{"log", "(", "x", ")"}]], TraditionalForm]],ExpressionUUID-> "a3329e89-1d56-4f52-9b31-559e8857e523"], " (and/or ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"\[Pi]", "(", "x", ")"}], "x"], TraditionalForm]],ExpressionUUID-> "8eabe64f-02f1-43b3-8483-ad74cd723e18"], " and ", Cell[BoxData[ FormBox[ FractionBox["1", RowBox[{"log", "(", "x", ")"}]], TraditionalForm]],ExpressionUUID-> "2e48fa45-ba23-4a01-8f45-16907cee3481"], ") for ", StyleBox["x", FontSlant->"Italic"], " up to ", Cell[BoxData[ FormBox[ SuperscriptBox["10", "5"], TraditionalForm]],ExpressionUUID-> "1e6df6bc-b047-4f62-a16c-30cfd472da19"], ", ", Cell[BoxData[ FormBox[ SuperscriptBox["10", "6"], TraditionalForm]],ExpressionUUID-> "9e65546f-9fca-48f5-9782-c6e186a97c1a"], " , ", Cell[BoxData[ FormBox[ SuperscriptBox["10", RowBox[{"7", " "}]], TraditionalForm]],ExpressionUUID-> "a5fb3ff2-6177-4c47-ae05-c715f7259d03"], ", ", Cell[BoxData[ FormBox[ SuperscriptBox["10", "8"], TraditionalForm]],ExpressionUUID-> "dbfafc46-168f-43f0-83e9-9436185af75a"], ", or even more if you can handle it. ", "(Taking ", StyleBox["x", FontSlant->"Italic"], " reals speeds up the computation). Is there a convergence?\n", "\n- In fact, asymptotic means that ", Cell[BoxData[ FormBox[ RowBox[{"\[Pi]", "(", "x", ")"}], TraditionalForm]],ExpressionUUID-> "03949ce9-809f-4a8c-969b-3efe9d0e0de9"], "/", Cell[BoxData[ FormBox[ FractionBox["x", RowBox[{"log", "(", "x", ")"}]], TraditionalForm]],ExpressionUUID-> "413947e0-186c-41b6-b39e-3df773e776f3"], " \[Rule] 1. Make plots of this. Is the convergence clear?" }], "Text", CellChangeTimes->{{3.447484168796875*^9, 3.44748424740625*^9}, { 3.447484302515625*^9, 3.44748434090625*^9}, {3.44748440853125*^9, 3.447484408640625*^9}, {3.4474844545*^9, 3.447484505296875*^9}, { 3.44748467215625*^9, 3.447484709484375*^9}, {3.447485266953125*^9, 3.447485347484375*^9}, {3.447485393234375*^9, 3.44748539540625*^9}, { 3.44748566625*^9, 3.44748566825*^9}, {3.44750038696875*^9, 3.44750044678125*^9}, {3.447502224125*^9, 3.44750223*^9}, { 3.447502313296875*^9, 3.447502380703125*^9}, {3.4475024481875*^9, 3.44750253865625*^9}, {3.447502580875*^9, 3.447502749203125*^9}, { 3.447502805640625*^9, 3.447503103859375*^9}, {3.447511491796875*^9, 3.447511491796875*^9}, {3.447511684796875*^9, 3.44751178165625*^9}, { 3.447585324046875*^9, 3.44758532784375*^9}, {3.45001595128125*^9, 3.4500160753125*^9}, {3.450016107109375*^9, 3.45001626103125*^9}, 3.450016396109375*^9, {3.793716245636414*^9, 3.7937162656415954`*^9}, { 3.7937163062572803`*^9, 3.7937163372728167`*^9}, {3.793717903390607*^9, 3.7937179562114754`*^9}, {3.793718064926051*^9, 3.7937182172990417`*^9}, { 3.79371916283041*^9, 3.7937191953987103`*^9}, {3.793719917093993*^9, 3.7937199747031097`*^9}, {3.7937204000419245`*^9, 3.793720418936866*^9}, { 3.7937204681563735`*^9, 3.793720469833965*^9}, {3.793720787418187*^9, 3.7937207880837507`*^9}, {3.79372088852099*^9, 3.7937208889251003`*^9}, { 3.907412840883844*^9, 3.9074128584218225`*^9}, {3.907412903978726*^9, 3.9074129780819716`*^9}, {3.907413653672807*^9, 3.9074137300451646`*^9}, { 3.9074153701803784`*^9, 3.9074153737333956`*^9}, {3.9074174060815954`*^9, 3.907417450672729*^9}, {3.9074175447359056`*^9, 3.9074176589448824`*^9}, { 3.9074177338176827`*^9, 3.9074178870044165`*^9}, {3.907418040658271*^9, 3.907418151177305*^9}, {3.9074182273145795`*^9, 3.9074184349023*^9}, { 3.9074185286044283`*^9, 3.9074186708646235`*^9}, {3.9074187450071697`*^9, 3.9074187865462523`*^9}, {3.907418865650231*^9, 3.907418882685953*^9}, { 3.907418934921791*^9, 3.9074190644272213`*^9}, {3.9074191683959513`*^9, 3.9074192149573836`*^9}, {3.9074243747761507`*^9, 3.90742437861698*^9}, { 3.907424471431411*^9, 3.9074244914605985`*^9}, {3.9074245218649607`*^9, 3.9074246898909516`*^9}, {3.907424855955865*^9, 3.9074250927267933`*^9}, { 3.907426229393008*^9, 3.9074262727021074`*^9}}, Background->RGBColor[ 0., 1., 0.],ExpressionUUID->"01f6f1c8-1c0c-41b1-9c8e-d2bd8ada30ee"], Cell[TextData[{ "13. Find the solutions of the system\n ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["x", "2"], "-", SuperscriptBox["y", "3"]}], TraditionalForm]],ExpressionUUID-> "f6066c48-ec45-4a2c-b738-2cc40c9e0673"], "=0 , xy ", Cell[BoxData[ FormBox[ RowBox[{"+", SuperscriptBox["x", "2"]}], TraditionalForm]],ExpressionUUID-> "539d06f6-c20b-462a-bdf6-8f02d528df2c"], "=0\nThen plot the two curves ", Cell[BoxData[ RowBox[{ SuperscriptBox["x", "2"], "-", SuperscriptBox["y", "3"]}]], "Input", CellChangeTimes->{{3.9074981133919225`*^9, 3.907498132702726*^9}}, ExpressionUUID->"8b1cf5c6-b76b-4fc6-a6f7-33ceb86f7de6"], " = 0, xy ", Cell[BoxData[ FormBox[ RowBox[{"+", SuperscriptBox["x", "2"]}], TraditionalForm]],ExpressionUUID-> "cd47fe97-b8b4-4522-87ba-4fc9efef19bf"], "=0 so as to show the solutions of the system as intersection points of the \ curves." }], "Text", CellChangeTimes->{{3.447484168796875*^9, 3.44748424740625*^9}, { 3.447484302515625*^9, 3.44748434090625*^9}, {3.44748440853125*^9, 3.447484408640625*^9}, {3.4474844545*^9, 3.447484505296875*^9}, { 3.44748467215625*^9, 3.447484709484375*^9}, {3.447485266953125*^9, 3.447485347484375*^9}, {3.447485393234375*^9, 3.44748539540625*^9}, { 3.44748566625*^9, 3.44748566825*^9}, {3.44750038696875*^9, 3.44750044678125*^9}, {3.447502224125*^9, 3.44750223*^9}, { 3.447502313296875*^9, 3.447502380703125*^9}, {3.4475024481875*^9, 3.44750253865625*^9}, {3.447502580875*^9, 3.447502749203125*^9}, { 3.447502805640625*^9, 3.447503103859375*^9}, {3.447511491796875*^9, 3.447511491796875*^9}, {3.447511684796875*^9, 3.44751178165625*^9}, { 3.447585324046875*^9, 3.44758532784375*^9}, {3.45001595128125*^9, 3.4500160753125*^9}, {3.450016107109375*^9, 3.45001626103125*^9}, 3.450016396109375*^9, {3.793716245636414*^9, 3.7937162656415954`*^9}, { 3.7937163062572803`*^9, 3.7937163372728167`*^9}, {3.793717903390607*^9, 3.7937179562114754`*^9}, {3.793718064926051*^9, 3.7937182172990417`*^9}, { 3.79371916283041*^9, 3.7937191953987103`*^9}, {3.793719917093993*^9, 3.7937199747031097`*^9}, {3.7937204000419245`*^9, 3.793720418936866*^9}, { 3.7937204681563735`*^9, 3.793720469833965*^9}, {3.793720787418187*^9, 3.7937207880837507`*^9}, {3.79372088852099*^9, 3.7937208889251003`*^9}, { 3.907412840883844*^9, 3.9074128584218225`*^9}, {3.907412903978726*^9, 3.9074129780819716`*^9}, {3.907413653672807*^9, 3.9074137300451646`*^9}, { 3.9074153701803784`*^9, 3.9074153737333956`*^9}, {3.9074174060815954`*^9, 3.907417450672729*^9}, {3.9074175447359056`*^9, 3.9074176589448824`*^9}, { 3.9074177338176827`*^9, 3.9074178870044165`*^9}, {3.907418040658271*^9, 3.907418151177305*^9}, {3.9074182273145795`*^9, 3.9074184349023*^9}, { 3.9074185286044283`*^9, 3.9074186708646235`*^9}, {3.9074187450071697`*^9, 3.9074187865462523`*^9}, {3.907418865650231*^9, 3.907418882685953*^9}, { 3.907418934921791*^9, 3.9074190644272213`*^9}, {3.9074191683959513`*^9, 3.9074192149573836`*^9}, {3.9074243747761507`*^9, 3.90742437861698*^9}, { 3.907424471431411*^9, 3.9074244914605985`*^9}, {3.9074245218649607`*^9, 3.9074246898909516`*^9}, {3.907424855955865*^9, 3.9074250927267933`*^9}, { 3.907426229393008*^9, 3.9074262727021074`*^9}, {3.9074964525676413`*^9, 3.90749657475708*^9}, {3.907496759549351*^9, 3.907496781997898*^9}, { 3.907497589331483*^9, 3.9074975999481196`*^9}, {3.907498237075573*^9, 3.9074983994263096`*^9}}, Background->RGBColor[ 0., 1., 0.],ExpressionUUID->"e069fa25-1b3b-4acc-b07b-73a9a34ceabb"], Cell[TextData[{ "14. Consider the function\n f: ", Cell[BoxData[ FormBox[ SuperscriptBox[ TemplateBox[{}, "Reals"], "2"], TraditionalForm]],ExpressionUUID-> "dbdcd75c-8565-4827-bae6-ae103cc1f363"], "\[Rule]", Cell[BoxData[ FormBox[ TemplateBox[{}, "Reals"], TraditionalForm]],ExpressionUUID-> "ff6dc432-7fa4-4ab3-ac50-5b7abad5aaff"], " , f(x,y) = ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "+", "y"}], ")"}], "2"], "-", "xy", "+", "1"}], TraditionalForm]],ExpressionUUID->"2ff63812-ec20-460b-a2ff-f636fa80538d"], " \n1. Find its critical points [The differential is D[ f[x,y] , \ {{x,y}} ]\n2. Plot the graph of f in a region that contains all critical \ points \n3. Superpose to the graph of f a point in each of its critical point \ \n [If the critical point has coordinates (x,y), create a graphics 3d \ object that represent a point (of a suitable PontSize) located at the point \ of coordinates\n (x,y,f(x,y)).\n Then, superpose it to the Plot of f \ with the Option ", StyleBox[" Epilog", FontWeight->"Bold"], " ", StyleBox["\[Rule] Graphics3D[ { PointSize[...], Point[{....}}]", FontWeight->"Bold"], "\n4. Investigate whether the fixd points are minima, maxima or saddle \ points [The Hessian is D[ f(x,y), { {x,y} , 2 } ]" }], "Text", CellChangeTimes->{{3.447484168796875*^9, 3.44748424740625*^9}, { 3.447484302515625*^9, 3.44748434090625*^9}, {3.44748440853125*^9, 3.447484408640625*^9}, {3.4474844545*^9, 3.447484505296875*^9}, { 3.44748467215625*^9, 3.447484709484375*^9}, {3.447485266953125*^9, 3.447485347484375*^9}, {3.447485393234375*^9, 3.44748539540625*^9}, { 3.44748566625*^9, 3.44748566825*^9}, {3.44750038696875*^9, 3.44750044678125*^9}, {3.447502224125*^9, 3.44750223*^9}, { 3.447502313296875*^9, 3.447502380703125*^9}, {3.4475024481875*^9, 3.44750253865625*^9}, {3.447502580875*^9, 3.447502749203125*^9}, { 3.447502805640625*^9, 3.447503103859375*^9}, {3.447511491796875*^9, 3.447511491796875*^9}, {3.447511684796875*^9, 3.44751178165625*^9}, { 3.447585324046875*^9, 3.44758532784375*^9}, {3.45001595128125*^9, 3.4500160753125*^9}, {3.450016107109375*^9, 3.45001626103125*^9}, 3.450016396109375*^9, {3.793716245636414*^9, 3.7937162656415954`*^9}, { 3.7937163062572803`*^9, 3.7937163372728167`*^9}, {3.793717903390607*^9, 3.7937179562114754`*^9}, {3.793718064926051*^9, 3.7937182172990417`*^9}, { 3.79371916283041*^9, 3.7937191953987103`*^9}, {3.793719917093993*^9, 3.7937199747031097`*^9}, {3.7937204000419245`*^9, 3.793720418936866*^9}, { 3.7937204681563735`*^9, 3.793720469833965*^9}, {3.793720787418187*^9, 3.7937207880837507`*^9}, {3.79372088852099*^9, 3.7937208889251003`*^9}, { 3.907412840883844*^9, 3.9074128584218225`*^9}, {3.907412903978726*^9, 3.9074129780819716`*^9}, {3.907413653672807*^9, 3.9074137300451646`*^9}, { 3.9074153701803784`*^9, 3.9074153737333956`*^9}, {3.9074174060815954`*^9, 3.907417450672729*^9}, {3.9074175447359056`*^9, 3.9074176589448824`*^9}, { 3.9074177338176827`*^9, 3.9074178870044165`*^9}, {3.907418040658271*^9, 3.907418151177305*^9}, {3.9074182273145795`*^9, 3.9074184349023*^9}, { 3.9074185286044283`*^9, 3.9074186708646235`*^9}, {3.9074187450071697`*^9, 3.9074187865462523`*^9}, {3.907418865650231*^9, 3.907418882685953*^9}, { 3.907418934921791*^9, 3.9074190644272213`*^9}, {3.9074191683959513`*^9, 3.9074192149573836`*^9}, {3.9074243747761507`*^9, 3.90742437861698*^9}, { 3.907424471431411*^9, 3.9074244914605985`*^9}, {3.9074245218649607`*^9, 3.9074246898909516`*^9}, {3.907424855955865*^9, 3.9074250927267933`*^9}, { 3.907426229393008*^9, 3.9074262727021074`*^9}, {3.9074964525676413`*^9, 3.90749657475708*^9}, {3.907496759549351*^9, 3.907496781997898*^9}, { 3.9074984391164675`*^9, 3.907498439850802*^9}, {3.9074988938198333`*^9, 3.907498899597869*^9}, {3.907499631964731*^9, 3.9075000858489122`*^9}, { 3.907506453615024*^9, 3.9075065462525015`*^9}, {3.9075066722825303`*^9, 3.907506691439015*^9}}, Background->RGBColor[ 0., 1., 0.],ExpressionUUID->"323ee36f-ea57-4a24-8699-d450e4be1aff"], Cell[TextData[{ "14. Consider the function\n f: ", Cell[BoxData[ FormBox[ SuperscriptBox[ TemplateBox[{}, "Reals"], "2"], TraditionalForm]],ExpressionUUID-> "7ca65954-d2ff-4c72-9637-fb48dbb0d58f"], "\[Rule]", Cell[BoxData[ FormBox[ TemplateBox[{}, "Reals"], TraditionalForm]],ExpressionUUID-> "26ce4b12-fb98-43c2-96c1-89e2ba6dbf30"], " , f(x,y) = ", Cell[BoxData[ SqrtBox[ RowBox[{ SuperscriptBox["x", "4"], "+", RowBox[{ SuperscriptBox["x", "2"], " ", SuperscriptBox["y", "2"]}], "+", SuperscriptBox["y", "4"], "-", "x", " ", "+", "1"}]]], "Input", CellChangeTimes->{{3.907498599005662*^9, 3.90749859936453*^9}, { 3.9074986993860817`*^9, 3.907498699793854*^9}, {3.9074987846111307`*^9, 3.9074987853222237`*^9}},ExpressionUUID-> "b839b773-d754-4698-9985-72bfc1e18f48"], "\n1. Determine the region of the plane where it is defined and plot it.\n2. \ Plot its graph.\n2. Find its critical points and investigate if they are \ minima, maxima or saddle points." }], "Text", CellChangeTimes->{{3.447484168796875*^9, 3.44748424740625*^9}, { 3.447484302515625*^9, 3.44748434090625*^9}, {3.44748440853125*^9, 3.447484408640625*^9}, {3.4474844545*^9, 3.447484505296875*^9}, { 3.44748467215625*^9, 3.447484709484375*^9}, {3.447485266953125*^9, 3.447485347484375*^9}, {3.447485393234375*^9, 3.44748539540625*^9}, { 3.44748566625*^9, 3.44748566825*^9}, {3.44750038696875*^9, 3.44750044678125*^9}, {3.447502224125*^9, 3.44750223*^9}, { 3.447502313296875*^9, 3.447502380703125*^9}, {3.4475024481875*^9, 3.44750253865625*^9}, {3.447502580875*^9, 3.447502749203125*^9}, { 3.447502805640625*^9, 3.447503103859375*^9}, {3.447511491796875*^9, 3.447511491796875*^9}, {3.447511684796875*^9, 3.44751178165625*^9}, { 3.447585324046875*^9, 3.44758532784375*^9}, {3.45001595128125*^9, 3.4500160753125*^9}, {3.450016107109375*^9, 3.45001626103125*^9}, 3.450016396109375*^9, {3.793716245636414*^9, 3.7937162656415954`*^9}, { 3.7937163062572803`*^9, 3.7937163372728167`*^9}, {3.793717903390607*^9, 3.7937179562114754`*^9}, {3.793718064926051*^9, 3.7937182172990417`*^9}, { 3.79371916283041*^9, 3.7937191953987103`*^9}, {3.793719917093993*^9, 3.7937199747031097`*^9}, {3.7937204000419245`*^9, 3.793720418936866*^9}, { 3.7937204681563735`*^9, 3.793720469833965*^9}, {3.793720787418187*^9, 3.7937207880837507`*^9}, {3.79372088852099*^9, 3.7937208889251003`*^9}, { 3.907412840883844*^9, 3.9074128584218225`*^9}, {3.907412903978726*^9, 3.9074129780819716`*^9}, {3.907413653672807*^9, 3.9074137300451646`*^9}, { 3.9074153701803784`*^9, 3.9074153737333956`*^9}, {3.9074174060815954`*^9, 3.907417450672729*^9}, {3.9074175447359056`*^9, 3.9074176589448824`*^9}, { 3.9074177338176827`*^9, 3.9074178870044165`*^9}, {3.907418040658271*^9, 3.907418151177305*^9}, {3.9074182273145795`*^9, 3.9074184349023*^9}, { 3.9074185286044283`*^9, 3.9074186708646235`*^9}, {3.9074187450071697`*^9, 3.9074187865462523`*^9}, {3.907418865650231*^9, 3.907418882685953*^9}, { 3.907418934921791*^9, 3.9074190644272213`*^9}, {3.9074191683959513`*^9, 3.9074192149573836`*^9}, {3.9074243747761507`*^9, 3.90742437861698*^9}, { 3.907424471431411*^9, 3.9074244914605985`*^9}, {3.9074245218649607`*^9, 3.9074246898909516`*^9}, {3.907424855955865*^9, 3.9074250927267933`*^9}, { 3.907426229393008*^9, 3.9074262727021074`*^9}, {3.9074964525676413`*^9, 3.90749657475708*^9}, {3.907496759549351*^9, 3.907496781997898*^9}, { 3.9074984391164675`*^9, 3.907498439850802*^9}, {3.9074988938198333`*^9, 3.907498899597869*^9}, {3.907506710297643*^9, 3.9075068404851837`*^9}, { 3.907506884462335*^9, 3.9075068963429565`*^9}, {3.9075069274882174`*^9, 3.907506927675955*^9}}, Background->RGBColor[ 0., 1., 0.],ExpressionUUID->"2bf6b894-da75-4e1d-bada-a7c70abf7266"] }, Open ]] }, WindowSize->{1141.2, 587.4}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, ShowGroupOpener->{"InsideFrame", "Inline"}, Magnification:>1.05 Inherited, FrontEndVersion->"13.0 for Microsoft Windows (64-bit) (February 4, 2022)", StyleDefinitions->"Default.nb", ExpressionUUID->"e74bcca4-1a77-4ddb-a3e1-a605d979e1d0" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[558, 20, 1544, 24, 81, "Text",ExpressionUUID->"b016031e-6c8e-4a66-ac65-1747af6335ee"], Cell[CellGroupData[{ Cell[2127, 48, 422, 6, 102, "Title",ExpressionUUID->"f729008c-babd-4da4-bc56-be901629beba"], Cell[2552, 56, 488, 9, 53, "Text",ExpressionUUID->"60ed2545-9357-4dd9-9a2f-7ac787a18d14"], Cell[3043, 67, 661, 14, 54, "Text",ExpressionUUID->"64e41634-1614-4033-8581-bd92fcdf0616"], Cell[3707, 83, 685, 14, 53, "Text",ExpressionUUID->"f994dc9a-238f-4cb5-b156-e4f1eaa9a8d4"], Cell[4395, 99, 1717, 34, 156, "Text",ExpressionUUID->"6827507b-dd8b-4b01-b59f-dc593c3b7887"], Cell[6115, 135, 3209, 94, 187, "Text",ExpressionUUID->"385a656a-334e-4f7b-bfcb-d56e600d5e61"], Cell[9327, 231, 1018, 20, 77, "Text",ExpressionUUID->"46d707c5-5062-4896-8900-bf26ee63f539"], Cell[10348, 253, 2118, 51, 151, "Text",ExpressionUUID->"2e3bcb28-f8a8-49cd-b557-d36ec4bdb18c"], Cell[12469, 306, 978, 17, 101, "Text",ExpressionUUID->"a5f7de18-7674-4454-9e91-07cd96288f80"], Cell[13450, 325, 3796, 89, 202, "Text",ExpressionUUID->"26aa55ac-ca53-464d-9fb1-1dca6d33fed6"], Cell[17249, 416, 25718, 428, 254, "Text",ExpressionUUID->"b5fb22c0-d1e8-4c7e-8968-38000943ae88"], Cell[42970, 846, 2052, 35, 101, "Text",ExpressionUUID->"edf53c7e-f392-403c-8d41-e2875e7eabf3"], Cell[45025, 883, 4934, 112, 218, "Text",ExpressionUUID->"01f6f1c8-1c0c-41b1-9c8e-d2bd8ada30ee"], Cell[49962, 997, 3652, 67, 101, "Text",ExpressionUUID->"e069fa25-1b3b-4acc-b07b-73a9a34ceabb"], Cell[53617, 1066, 4231, 76, 247, "Text",ExpressionUUID->"323ee36f-ea57-4a24-8699-d450e4be1aff"], Cell[57851, 1144, 3907, 70, 152, "Text",ExpressionUUID->"2bf6b894-da75-4e1d-bada-a7c70abf7266"] }, Open ]] } ] *)